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I believe you are thinking it backwards. Let's say that your non-redirected (existing) URL is /foo/bar/baz/bat.html and your redirected (new) URL is foobar. When you share the short link foobar what happens is that the social media site (e.g. Facebook or Twitter) tries to access it. First it finds Admin Tools' redirection which tells it to look at the /foo/bar/baz/bat.html URL instead. This is where Admin Tools' responsibility begins and ends.

The /foo/bar/baz/bat.html URL is rendered using a different component. Which one? I have no idea, you never told me :) If it's Joomla's core Articles you can probably use an extension like Phoca Open Graph. That's what I use on my personal blog. Social media pick up the Open Graph tags and display images and metadata about your content in the sharing preview.